The lattice size $\operatorname{ls_\Delta}(P)$ of a lattice polygon $P$ with respect to the standard simplex $\Delta$ was introduced and studied by Castryck and Cools in the context of simplification of the defining equation of an algebraic curve. Earlier, Schicho provided an “onion skins” algorithm for mapping a lattice polygon $P$ into a small integer multiple of the standard simplex, based on passing successively to the convex hull of the interior lattice points of $P$. Castryck and Cools showed that this algorithm computes the lattice size of $P$. In this paper we show that for a plane convex body $P$ a reduced basis of $\mathbb Z^2$ computes the lattice size. This provides a lattice reduction algorithm for computing the lattice size, which works for any convex body $P\subset\mathbb R^2$ and outperforms the “onion skins” algorithm in the case when $P$ is a lattice polygon.

This paper is concerned with the minimum distance computation for higher dimensional toric codes defined by lattice polytopes in $\mathbb{R}^n$. We show that the minimum distance is multiplicative with respect to taking the product of polytopes, and behaves in a simple way when one builds a k-dilate of a pyramid over a polytope. This allows us to construct a large class of examples of higher dimensional toric codes where we can compute the minimum distance explicitly.

In this paper we prove new lower bounds for the minimum distance of a toric surface code $\mathcal{C}_P$ defined by a convex lattice polygon $P\subset\mathbb{R}^2$. The bounds involve a geometric invariant $L(P)$, called the full Minkowski length of P. We also show how to compute $L(P)$ in polynomial time in the number of lattice points in P.